[Paper Review] Irreducible representations of untwisted affine Kac-Moody algebras

Xiangqian Guo, Kaiming Zhao|arXiv (Cornell University)|May 17, 2013

Algebraic structures and combinatorial models11 references13 citations

TL;DR

This paper constructs a new class of irreducible modules over untwisted affine Kac-Moody algebras by combining loop modules and modules with special properties, generalizing highest weight and Whittaker modules. It provides a complete classification of irreducible modules where the action of each root vector in the positive part is locally finite, and establishes a necessary and sufficient condition for the simplicity of tensor products of integrable highest weight and loop modules.

ABSTRACT

In this paper we construct a class of new irreducible modules over untwisted affine Kac-Moody algebras $\widetilde{\mathfrak{g}}$, generalizing and including both highest weight modules and Whittaker modules. These modules allow us to obtain a complete classification of irreducible $\widetilde{\mathfrak{g}}$-modules on which the action of each root vector in $\widetilde{\mathfrak{n}}_+$ is locally finite, where $\widetilde{\mathfrak{n}}_+$ is the locally nilpotent subalgebra (or positive part) of $\widetilde{\mathfrak{g}}$. The necessary and sufficient conditions for two such irreducible $\widetilde{\mathfrak{g}}$-modules to be isomorphic are also determined. In the second part of the paper, we use the "shifting technique" to obtain a necessary and sufficient condition for the tensor product of irreducible integrable loop $\widetilde{\mathfrak{g}}$-modules and irreducible integrable highest weight $\widetilde{\mathfrak{g}}$-modules to be simple. This tensor product problem was originally studied by Chari and Pressley 28 years ago.

Motivation & Objective

• To construct new irreducible modules over untwisted affine Kac-Moody algebras that generalize both highest weight and Whittaker modules.
• To classify all irreducible modules on which the action of each root vector in the positive part is locally finite.
• To determine necessary and sufficient conditions for the irreducibility of tensor products of integrable highest weight and loop modules.
• To extend the classification of irreducible modules beyond finite-dimensional weight spaces to include infinite-dimensional cases.
• To resolve a long-standing problem posed by Chari and Pressley on the simplicity of such tensor products, 28 years after initial study.

Proposed method

• Constructs new irreducible modules via induction from the tensor product of an irreducible evaluation module for $\mathfrak{g} \otimes \mathbb{C}[t,t^{-1}]$ and a module $L$ with a special property (e.g., highest weight or Whittaker module).
• Uses the induced module $\mathrm{Ind}_{\widehat{\mathfrak{g}}}^{\widetilde{\mathfrak{g}}}(E \otimes L)$ to generate irreducible $\widetilde{\mathfrak{g}}$-modules.
• Applies the 'shifting technique' to analyze the structure of tensor products of integrable highest weight and loop modules.
• Analyzes the action of $e \otimes t^{-1}$ on weight spaces to determine irreducibility conditions via dimension comparisons in specific weight spaces.
• Employs explicit computations of $ (e \otimes t^{-1})^j (V(\epsilon) \otimes V(2\epsilon)) $ to evaluate when $ W^{(j)} = W $, a key criterion for irreducibility.
• Uses representation-theoretic tools such as Verma-type modules, Borel subalgebras from non-standard root system partitions, and weight space decompositions.

Experimental results

Research questions

• RQ1 What is the complete classification of irreducible $\widetilde{\mathfrak{g}}$-modules on which the action of each root vector in $\widetilde{\mathfrak{n}}_+$ is locally finite?
• RQ2 Under what conditions is the tensor product of an irreducible integrable highest weight module and an irreducible integrable loop module simple?
• RQ3 Can the irreducibility of induced modules constructed from evaluation modules and special $\widehat{\mathfrak{g}}$-modules be determined via explicit weight space dimension comparisons?
• RQ4 How does the function $\kappa(\boldsymbol{\lambda}, \mathbb{a}, \Lambda)$, which controls irreducibility, depend on the parameters $\boldsymbol{\lambda}, \mathbb{a}, \Lambda$?
• RQ5 Are there explicit formulas for $\kappa(\boldsymbol{\lambda}, \mathbb{a}, \Lambda)$ in terms of the highest weights and parameters $a_i$?

Key findings

• The paper provides a complete classification of irreducible $\widetilde{\mathfrak{g}}$-modules where the action of each root vector in $\widetilde{\mathfrak{n}}_+$ is locally finite, including highest weight modules, Whittaker modules, and new non-weight modules.
• For $\mathfrak{sl}_2^{(1)}$, the irreducibility of $E((\epsilon,2\epsilon);(a_1,a_2);b) \otimes \widetilde{V}(i\epsilon + k\Lambda_0)$ holds if and only if $W^{(k-i+1)} = W$, where $W^{(j)}$ is the span of $ (e \otimes t^{-1})^j (V(\epsilon) \otimes V(2\epsilon)) \otimes u $.
• For $i=0$, $\kappa((\epsilon,2\epsilon);(a_1,a_2);0) = 2$ for all distinct $a_1, a_2 \in \mathbb{C}^*$, meaning the tensor product is irreducible only when $k=1$.
• For $i=1$, $\kappa((\epsilon,2\epsilon);(a_1,a_2);\epsilon) = 2$, so irreducibility holds only for $k=1$.
• For $i \geq 2$, $\kappa((\epsilon,2\epsilon);(a_1,a_2);i\epsilon) = i$, indicating that irreducibility fails for $k \geq i$.
• The function $\kappa$ is independent of the parameters $a_1$ and $a_2$, showing a surprising invariance in the irreducibility condition.